Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C27 |
| direct product | C53×#20964 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | C2×#120 |
| commutator | C23 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| socle | C2×C53 | C2×#386 |
| prime radical | C53×#3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C23×C4×C53 ◁ G | |
1700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,10,6,9,18,23)(3,7,13,21,12,20,27,31)(4,14,16,24,15,8,17,22)(11,19,26,30,25,29,28,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,5,13,6,12,18,27)(2,7,10,21,9,20,23,31)(4,11,17,28,15,25,16,26)(8,19,24,32,14,29,22,30);(1,4)(2,8)(3,11)(5,17)(6,15)(7,19)(9,14)(10,24)(12,25)(13,28)(16,18)(20,29)(21,32)(22,23)(26,27)(30,31);(2,9)(7,20)(8,14)(10,23)(19,29)(21,31)(22,24)(30,32)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 13952);